JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, 5659–5672, doi:10.1002/jgra.50545, 2013
Energy transfer and flow in the solar wind-magnetosphere-ionosphere
system: A new coupling function
P. Tenfjord1,2 and N. Østgaard1,2
Received 23 April 2013; revised 5 August 2013; accepted 31 August 2013; published 16 September 2013.
[1] In this paper we describe and quantify the energy transfer, flow, and distribution.
Our high-resolution data set covers 13 years of OMNI, SuperMAG, and Kyoto data. We
employ what we consider to be the best estimates for energy sinks and relate these to
SuperMAG indices for better coverage and spatial resolution. For the energy input, we
have used the method of dimensional analysis that is presented in unit power and makes it
appropriate for energy analysis. A cross-correlation analysis parameterizes the
magnetospheric response on the solar wind parameters during a wide range of conditions,
ranging from substorms and storms up to a decade. The determined functional form is
then evaluated and scaled using superposed epoch analysis of geomagnetic storms,
revealing that the effective area of interaction cannot be considered static. Instead, we
present a dynamic area which depends to the first order on the cube of the interplanetary
magnetic field Bz component. Also, we find that for longer time periods, this area must be
increased compared to the area used for geomagnetic storms. We argue that some of the
terms in the energy coupling function are contributory to describing magnetosheath
conditions and discuss how our coupling function can be related to Maxwell stress
components. Also, we quantify the relative importance of the different energy sinks
during substorms, geomagnetic storms, and long time series and present the coupling
efficiency of the solar wind. Our energy coupling functions are compared with the parameter and perform better for almost any event.
Citation: Tenfjord, P., and N. Østgaard (2013), Energy transfer and flow in the solar wind-magnetosphere-ionosphere system: A
new coupling function, J. Geophys. Res. Space Physics, 118, 5659–5672, doi:10.1002/jgra.50545.
1. Introduction
[2] The solar wind is the ultimate source of energy and
is responsible for virtually all the magnetospheric dynamics.
Describing and quantifying the solar wind energy transfer to
the Earth’s magnetosphere is one of the fundamental questions in space physics. It is well accepted that the energy
transfer is closely related to the Bz component of the interplanetary magnetic field (IMF), and through a large-scale
reconnection process on the dayside, a small fraction of the
available solar wind energy is able to penetrate our magnetosphere [Dungey, 1961]. However, this tiny fraction of energy
fuels almost all the dynamical processes in the near-Earth
space. An enhanced understanding of how the solar wind
energy couples with the magnetosphere is an essential part of
creating a reliable prediction of the space weather. Our modern society has become increasingly dependent on reliable
1
Birkeland Centre for Space Science, Bergen, Norway.
Department of Physics and Technology, University of Bergen, Bergen,
Norway.
2
Corresponding author: P. Tenfjord, Department of Physics and Technology, University of Bergen, Allegt 55, Bergen NO-5007, Norway.
([email protected])
©2013. American Geophysical Union. All Rights Reserved.
2169-9380/13/10.1002/jgra.50545
technology, such as communication, navigation, and electrical power systems, which can be vulnerable to the solar
winds influence.
[3] The most important forms of ionospheric and magnetospheric energy dissipation are Joule heating of the
ionosphere (UJ ), ring current injection (UR ), and particle
precipitation (UA ) [Akasofu, 1981; Gonzalez and Gonzalez,
1984; Østgaard et al., 2002a; Turner et al., 2009]. The
energy deposited into these sinks can be estimated using
ground-based magnetometer data [Dessler and Parker,
1959; Sckopke, 1966; Akasofu, 1981; Ahn et al., 1983].
Energy dissipated through minor ionospheric channels, e.g.,
ion outflow and auroral kilometric radiation, and magnetospheric energy dissipation processes such as plasmoid ejection and plasma sheet heating are neglected, because they
are small or not part of the closed magnetospheric system
of interest (e.g., plasmoid) [Koskinen and Tanskanen, 2002;
Baker et al., 1997; Knipp et al., 1998; Ieda et al., 1998].
There is no direct method of monitoring the energy transferred to the magnetosphere by the solar wind. The local
energy conversion across the magnetopause has been estimated with Cluster; however, this is restricted both in spatial and temporal resolution [see Rosenqvist et al., 2006,
2008; Anekallu et al., 2011]. Instead, one estimates the
energy needed to drive the magnetospheric system using
ionospheric and magnetospheric indices as proxies. The
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TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
so-called coupling function is then related to upstream
solar wind parameters and evaluated in terms of the correlation with estimated energy dissipative processes in the
ionosphere and inner magnetosphere.
[4] The most widely used energy coupling function is
the parameter by Akasofu and Perreault [1978]. The parameter is a semiempirical parameter representing the
energy input rate (W, J/s). The parameter was parameterized by assuming that the energy required to power the
magnetosphere-ionosphere system is provided by the solar
wind. The required energy is a measure of the total rate
of energy dissipation in the inner magnetosphere and ionosphere. The parameter is given in SI units as [Koskinen and
Tanskanen, 2002]
[W] =
4 2 4
VB sin ( /2)l20
0
(1)
where V,B, , and l0 are the solar wind velocity, total magnetic field, clock angle, and the effective area of interaction,
respectively. The clock angle is the angle between the IMF
vector projected into the GSM Y – Z plane and the Z axis,
positive in the clockwise direction as seen from the Sun. The
parameter is often interpreted as to be derived from the
Poynting flux.
[5] A more general expression for the reconnection power
was presented by Gonzalez and Mozer [1974], where the
authors explained the energy transfer as the Joule heating
rate of the reconnection-electric field and the magnetopause
current. In a comparison by Gonzalez and Gonzalez [1984]
between the parameter and the more general expression
[Gonzalez and Mozer, 1974], the authors showed that for the
limit that involves equal geomagnetic and magnetosheath
magnetic field amplitudes, their functional form reduces to
that of the parameter.
[6] In recent years, several coupling functions have been
proposed by numerous authors (see Gonzalez et al. [1994] or
Newell et al. [2007] for a review), attempting to parameterize the magnetospheric response to solar wind driving more
accurately. Perhaps the main reason why the parameter is
the most widely used is that it quantifies the energy transfer
in terms of power (W), and the time-integrated values (J) can
be compared to estimated energy sinks in the ionosphere and
inner magnetosphere. Also, the parameter is to our knowledge the only energy coupling function which is scaled to
the magnetospheric energy sinks.
[7] Even though Akasofu [1981] made it clear that the parameter should be considered as a first approximation, it
has without doubt played a crucial part of our general understanding of solar wind-magnetosphere-ionospheric interaction, as will be outlined below. The parameter was
originally scaled to geomagnetic storm events; however, it
has been used to study a wide range of time scales and
conditions.
[8] Stamper et al. [1999] studied century-long coupling
between geomagnetic activity and solar wind in search for
mechanisms responsible for long-term increase in geomagnetic activity. Knipp et al. [1998] presented an event-specific
origin-to-end view of an geomagnetic storm event. While
Kallio et al. [2000] used the parameter to study the
loading-unloading processes during substorms, Tanskanen
et al. [2002] analyzed the substorm energy budget during low and high solar activities. Lu et al. [1998]
used the parameter to quantify the global magnetospheric energy deposition during a magnetic cloud event.
Palmroth et al. [2010] studied the magnetospheric feedback in the energy transfer process and, using the Grand
Unified Magnetosphere Ionosphere Coupling Simulation,
version 4 (GUMICS-4), found that the energy transfer can
continue even after the direct driving conditions turned unfavorable. This was termed as hysteresis effect. Koskinen and
Tanskanen [2002] addressed the underlaying physical foundation for the parameter and clarified how the parameter
should be interpreted and used.
[9] In the last 30 years, the data and our understanding
of the dissipative processes in the inner magnetosphere and
ionosphere have been greatly improved, and it has been generally accepted that the scaling factor, l0 , in the parameter
is somewhat low [Østgaard et al., 2002b; Koskinen and
Tanskanen, 2002].
[10] The purpose of this paper is to derive an energy
coupling function that is valid for a wide range of solar
wind conditions and time scales, ranging from substorms
and storms, up to a decade. We will require the function to
balance the energy sinks in the ionosphere and inner magnetosphere with the estimated magnitude much improved
since the parameter was derived. By letting the effective
area of energy transfer be a free parameter, we will optimize the coupling function to be valid for long time series.
This new and dynamic energy coupling function will then
be discussed in terms of Maxwell stress and magnetosheath
conditions in order to gain a better understanding of how
the flow of energy is transferred from the solar wind to the
magnetosphere.
2. Theoretical Background
2.1. Energy Source and Sinks
[11] In this section we introduce the energy source and
sinks in the solar wind-magnetosphere-ionosphere system.
2.1.1. Solar Wind Kinetic Energy
[12] The solar wind is the ultimate source of energy in the
system. The kinetic energy flux of the solar wind available
for the magnetosphere is
USW =
1 3
V A
2 x
(2)
where A is the cross-sectional area of the magnetosphere,
assuming cylindrical symmetric configuration, and Vx is the
x component of the solar wind velocity. Magnetic and thermal energies in the solar wind are usually 1 or 2 orders of
magnitude smaller than kinetic energy and are neglected for
this reason.
[13] To estimate A, we use the empirical model of the
near-Earth magnetotail geometry by Petrinec and Russell
[1996]. Using this model, we calculate the magnetotail
radius, RT , for a given nightside position of XGSM = –30 RE .
Entering this into equation (2), the available solar wind
kinetic energy flux is estimated.
2.1.2. Joule Heating
[14] Joule heating is caused by ionospheric currents which
heat the atmosphere and takes place through the Pedersen
currents associated with the closure of field aligned currents
in the resistive ionosphere [Koskinen and Tanskanen,
2002]. The mechanism can be recognized as Ohmic
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TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
dissipation: loss of electric energy when a current flows
through a resistive medium, where the work done on the
particles will increase their mechanical energy. Vasyliunas
[2005] argues that Joule heating is not primarily Ohmic or
Joule heating in the physical sense, rather simply frictional
heating from the relative motion of plasma and neutrals
[Song et al., 2009].
[15] Joule heating can be estimated using various methods
and techniques and has been studied using experimental data
[Ahn et al., 1983, 1989; Baumjohann and Kamide, 1984;
Richmond et al., 1990], large statistical studies [Foster et al.,
1983], assimilative data methods (assimilative mapping of
ionospheric electrodynamics (AMIE)) [Lu et al., 1998], and
global MHD simulations [Palmroth et al., 2004]. There is
a dispute in the community about the actual contribution
of Joule heating [Vasyliunas, 2005], and the different techniques give contradictory results [Palmroth et al., 2006].
Among the experimental results, the estimates differ by
approximately a factor 2, while the MHD results deviate
by more than a factor 10. A thorough assessment of Joule
heating can be found in Palmroth et al. [2005].
[16] In this paper we use results from experimental studies
and estimate the Joule heating rate using the auroral electrojet (AE) index as a proxy. Østgaard et al. [2002b] reviewed
several results, and we adapt their conclusion for the reasons
given there. In their paper the results from Ahn et al. [1989]
(summer) and Richmond et al. [1990] (winter) were used.
The Joule heating in both hemispheres during solstice is then
given by
UJ [GW] = 0.54AE + 1.8
(3)
We relate Joule heating rate to the SuperMAG indices
instead of AE, by determining an average value of the ratio
SuperMag electrojet (SME)/AE during several periods when
the Kyoto AE magnetometers were well located. We simply
use the ratio to rewrite equation (3) and express the Joule
heating rate as a function of SME:
UJ [GW] = 0.48SME
UA [GW] = 4.1 (5)
Østgaard et al. [2002a] used data derived from UV and Xray emission during seven isolated substorms verified using
DMSP data [Østgaard et al., 2001] and Kyoto Quicklook
indices (six to eight stations). During two of these substorms,
the stations available were poorly located. Therefore, only
five substorms were used in their analysis.
(6)
SML – 9
We acknowledge that the method used to rewrite the Joule
heating rate and particle precipitation to be functions of
SuperMAG may seem brutal; however, in our opinion, the
advantage of superior coverage and spatial resolution in
SuperMAG exceeds the rough estimate. UA may also have
a seasonal dependence which is unknown [see Barth et al.,
2004; Newell et al., 1996]. This will be discussed in
section 7.
2.1.4. Ring Current Injection
[20] Via the so-called Dessler-Parker-Schopke (DPS)
relation [Dessler and Parker, 1959; Sckopke, 1966], the Dst
index can be used to estimate the energy increase of the ring
current. During disturbed solar wind conditions, variations
in the solar wind pressure modulate currents flowing at the
dayside magnetopause [Gonzalez et al., 1994]. The magnetosphere is compressed by the sudden increase in solar wind
dynamic pressure and at the same time the eastward magnetopause current intensifies. This is observed as an increase
in the Dst index. However, this is not related to the ring
current and we therefore remove this effect by using the
pressure-corrected Dst index:
Dst* = Dst – H
(7)
By adopting the analysis by Østgaard et al. [2002b], we use
the following equation for the pressure correction [Gonzalez
et al., 1994; Prigancová and Feldstein, 1992]:
H [nT] = 5 105 (nT (J m–3 )–1/2 ) P1/2
f – 20 nT
(4)
We notice that the values of Joule heating have a seasonal
dependence [see Østgaard et al., 2002b, Table 2], so our
solstice values may not be valid for all seasons. Ahn et al.
[1983] suggested UJ = 0.23AE (one hemisphere) during
equinox; this implies a seasonal difference from solstice to
equinox of 15%. We will discuss how other choices of UJ
would affect our results in section 7.
2.1.3. Auroral Precipitation
[17] Particles with their mirror point below 100 km are
very likely to deposit their energy through ionization, aurora,
and heating, due to increased density of neutral particles
below this height. By deriving the electron energy deposition
(0.1–100 keV) from UV and X-ray emissions in the Northern Hemisphere, Østgaard et al. [2002a] found the relation
between AL index and energy deposited to be
UA [GW] = 4.4 AL1/2
QL – 7.6
[18] By using the same data set as Østgaard et al. [2002a],
with final Kyoto indices (12 stations) and seven substorms,
the same analysis has been performed and a slightly different relation, which is valid for ALfinal , is given as UA =
4.3AL1/2 – 9.
[19] We rewrite UA to be a function of the SuperMAG
lower (SML) index by the same method as used for Joule
heating:
p
(8)
where Pf is the solar wind dynamic pressure given in pascal.
UR [GW] = –4 104
@Dst* Dst*
+
@t
(9)
with the ring current decay rate given as = 4, 8 or 20 h
for Dst* < –50, < –30, and > –30 nT, respectively. We have
adopted the values from Lu et al. [1998] but will discuss the
effect of using a dynamic value as proposed by O’Brien
and McPherron [2000].
[21] We have used the SYM-H index which we consider
to be the equivalent high-resolution version of the Dst index
[Wanliss and Showalter, 2006]. The SuperMAG SMR index
[Newell and Gjerloev, 2012] was also considered; however, due to its generally higher values, this resulted in an
underestimate of the loss term in equation (9).
[22] The total energy dissipation rate is defined as the
sum of particle precipitation, Joule heating, and ring current
injection:
UT = UA + UJ + UR
(10)
2.2. Energy Coupling Between the Solar Wind
and Magnetosphere
[23] Electric field related coupling functions [e.g.,
Sonnerup, 1974; Burton et al., 1975; Kan and Lee, 1979;
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TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
Newell et al., 2007; Borovsky, 2008; Milan et al., 2012] generally express the reconnection electric field, or the rate at
which magnetic flux is opened at the magnetopause. From
an energy perspective, the reconnection rate is a measure
of field lines added to the magnetotail. The energy density
associated with a magnetic field is also the magnetic pressure given by B2 /20 [Pa, J/m3 ]. Thus, the product VB2 /20
represents the flow of magnetic energy, or rate at which the
energy contained in the field lines is transported through
the magnetopause (energy flux density through unit area).
The energy transfer rate of solar wind to the magnetosphere
is generally explained from an electrodynamical view, by
using the Poynting flux to relate the concept of energy to the
fields. The energy transfer is then described as the amount
of Poynting flux admitted into the magnetosphere. Thus,
the conservation of energy is established by means of the
Poynting theorem, which states that the power delivered into
the magnetospheric volume from electromagnetic fields is
equal to the Joule heating in the volume plus the increase in
the energy stored in the fields in the volume. In this view,
the energy transfer process can be expressed in terms of the
familiar laws of electric circuit theory. The dynamics can
then be understood in terms of Joule heating at the magnetopause (generator) [Gonzalez and Mozer, 1974], inductance
of the magnetic fields in the tail, and the ionosphere as the
resistor (load). Qualitatively, the entire high-latitude boundary is a generator (EE JE < 0) [Koskinen and Tanskanen,
2002; Yeh et al., 1981]. The Poynting vector can be written
E) B
E /0 = vE? B2 /0 . Describing the energy
as SE = (–Ev B
transfer via the Poynting flux is without doubt the most convenient way, it also allows the use of the familiar laws of
electrodynamics; however, EE and JE are derived from Ev and
E and do not control the dynamics [Parker, 2007]. ChanB
neling upstream Poynting flux from the solar wind to the
magnetosphere does not provide an adequate description of
the transfer processes, as the bow shock and magnetosphere
are actually sources of Poynting flux. Also, in the reference
frame of the plasma, the Poynting vector vanishes; the correct description of the energy transfer should be valid in any
frame of reference.
[24] It has become generally accepted that the correct
description of the energy transfer [Rosenqvist et al., 2006;
Koskinen and Tanskanen, 2002] is via the Maxwell magnetic stress tensor [Siscoe and Cummings, 1969; Siscoe and
Crooker, 1974]. The quantitative formulation was given by
Siscoe and Cummings [1969]. The general description is
based on the conservation of momentum. The conservation
of momentum equation, which is a consequence of Newton’s second law of motion, states that the rate of change
of momentum in a differential volume is equal to the total
force acting on it. In ideal magnetohydrodynamics (MHD),
the magnetic field is obliged to move with the plasma;
thus, the plasma experiences the magnetic stresses. The
MHD momentum equation can be written as [cf. Siscoe and
Siebert, 2006; Siscoe, 1972; Parker, 2007]
@vj
vi @vj @pij @Mij
+
–
=–
@t
@xi
@xi
@xi
or
(11)
I
Fj =
–Rij – pij + Mij ni d
(12)
where the divergence theorem has been used to convert
equation (11) to an area integral (equation (12)), is the
BB
B2
+ i 0j is the Maxwell magnetic
mass density, Mij = –ıij 2
0
stress tensor, Rij = vi vj is the Reynolds stress (momentum
flux density transported by the mean bulk velocity), and pij
is the thermal pressure tensor (momentum flux density transported by the thermal motions). The physical meaning of ij
could be referred to as the force acting in the jth direction on
a surface that is oriented perpendicular to the ith direction.
In equation (12), the right-hand side is the closed surface
integral over the momentum stress tensor. The left-hand
side represents the total acceleration of the mass, hence, the
force on the magnetosphere [Siscoe and Siebert, 2006]. The
forces acting perpendicular to the surface are called normal
stresses and forces acting tangential to the surface tangential
stresses. Siscoe [1972] argues that only tangential stresses
result in energy transfer, since a tangential force acts against
the solar wind flow. Thus, in the absence of tangential
forces at the tail boundary, the forces are always perpendicular to the solar wind flow and no energy is extracted.
Tangential stresses are associated with viscous (e.g., KelvinHelmholtz instability) and electromagnetic (electric and
magnetic) stresses. The momentum transport from thermal
motions is assumed to play no significant role [Vasyliunas
et al., 1982]. The Reynolds stress is also neglected; the
diagonal terms are neglected from our assumption that the
solar wind flows parallel to the tail boundary. The shear
components that arise from turbulent fluctuations on the
mean flow, which can be associated with eddy viscosity
[Borovsky and Funsten, 2003], are significant when studying viscous-related momentum transfer but here considered
small compared to electromagnetic stresses. Thus, considering only electromagnetic forces (we neglect electric stresses
because it is small [Siscoe, 1972]), the total force can be
expressed as the magnetic surface stress:
I
Fj =
I
Mij ni d =
1
0
1
Bi Bj – ıij B2 ni d
2
(13)
The diagonal terms in equation (13) represent magnetic pressure, and the off-diagonal elements are shears. We define
the shear stress in the Maxwell stress tensor (i ¤ j in
equation (13)) in cylindrical coordinates as
Ft =
1
Bn Bt
0
(14)
where Ft denotes the tangential force and Bn and Bt are the
normal and tangential components of the magnetic field at
the magnetopause. The power delivered to the field can then
be expressed as [cf. Siscoe and Cummings, 1969; Siscoe and
Crooker, 1974; Vasyliunas et al., 1982]
I
P=
Bn BEt VE
d
0
(15)
where VE is the plasma flow (assumed equal to upstream
solar wind velocity) and the integral is taken over the surface of the magnetotail. The main mechanism which results
in tangential magnetic stress on the tail boundary is magnetic reconnection. When the IMF merges with the terrestrial
magnetic field at the subsolar point, the stress in the field
lines attempts to straighten the field lines and accelerate the
plasma. Behind the terminator plane (approximately), the
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TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
1–˛ V3–2˛ B2˛
T 2
PV = V3 l2CF M–2˛
lCF F( )
A F( ) =
˛
0
(16)
where is the solar wind density, V the solar wind velocity,
and BT is the transverse
interplanetary magnetic field (IMF)
q
defined as BT = B2y + B2z . The Alfvén Mach number, based
on the transverse magnetic field, can be expressed as
MA =
v
=
vA
p
0 Pf
BT
[26] We have employed the OMNI data set for solar
wind measurements. OMNI (available from NASA’s Space
Physics Data Facility, http://omniweb.gsfc.nasa.gov/) is a
multispacecraft (ACE, Wind, Geotail, IMP 8) compilation
of near-Earth, high-resolution solar wind magnetic field and
plasma data, shifted to Earth’s bow shock nose using a
minimal variance technique [Weimer et al., 2003]. It also
contains geomagnetic activity indices (Kp, SYM-H, AE),
sunspot number, and additional computed parameters such
as plasma flow pressure, plasma beta, Alfvén Mach number,
and electric field [King and Papitashvili, 2005]. We use 5
min resolution and the data set covers a period from January
1997 to January 2010.
[27] In addition to the indices that are included in the
OMNI data, we also use SME and SML. These magnetic
indices are derived from the magnetometer data provided by
the SuperMAG collaborators [Gjerloev, 2012; Newell and
Gjerloev, 2011a, 2011b]. SuperMAG (http://supermag.uib.
no) is a worldwide collaboration of organizations and
national agencies that currently operate more than 300
ground-based magnetometers.
PV(α) [W (normalized)]
magnetic field slows down the solar wind plasma as the field
lines are stretched tailward. This region we define as the
transfer region, LT , and we consider the energy extraction to
be limited by the start of solar wind-retarding forces and the
intersection with the boundary of the energy flux line that
connects to the merging point in the neutral sheet [Siscoe and
Crooker, 1974], see Figure 1. This is approximately beyond
the cusp region to the position where the near-Earth neutral
line (NENL) is located.
[25] There are no direct method to measure the total
rate at which energy is extracted from the solar wind by
the magnetosphere and no obvious theoretical method to
calculate Bn and Bt (see equation (15)) from first principles (this will be discussed further in section 7). Rather,
we employ a semiempirical analysis using the dimensional
analysis introduced by Vasyliunas et al. [1982], intending
to express the dimension of a quantity as a product of
the basic fundamental dimensions. We cross-correlate solar
wind parameters to the estimated energy sinks and use the
constraints imposed by the dimensional analysis as a template to parameterize a general formula expressing the rate
of solar wind-magnetosphere energy transfer [Vasyliunas et
al., 1982]. The general expression for the rate of energy
transfer, involving only MHD flow and by assuming that
the Chapman-Ferraro length remains constant at the magnetopause [Gonzalez, 1991], can be expressed as [Vasyliunas
et al., 1982]
3. Instrumentation and Data Processing
UT [GW]
Figure 1. Sketch showing the magnetosphere with magnetic field lines in merging configuration. The energy flow
pattern is shown as blurry lines indicating energy transfer
from the tail boundary. The distance LT is measured from the
beginning of solar wind-retarding forces and limited by the
boundary where the energy flow line intersects the merging
point (NENL). Adopted from Siscoe and Crooker [1974].
where vA is the Alfvén velocity. The angular dependence,
F( ), represents the reconnection efficiency from zero, when
the IMF is parallel to the magnetopause field, to maximum as the IMF rotates to being antiparallel. A separate
analysis, not presented in the paper, showed that the correlation was not strongly dependent on the exponent used
in the term sinˇ ( /2). Overall we found, based on trials,
that F( ) = sin4 ( /2) performed better than the alternatives
(ˇ = 1, 2, 3, 5). This will be discussed in section 7. The
physical interpretation of ˇ = 4 is that only the dawn-dusk
component of the reconnection electric field contributes to
the energy transfer [Gonzalez and Mozer, 1974; Kan et al.,
1980; Gonzalez and Gonzalez, 1981]. In comparison, Newell
et al. [2008] used ˇ = 8/3 while Milan et al. [2012] found
the best correlation when ˇ = 4.5.
(17)
1
α = 0.5
α=1
0.8
0.6
0.4
0.2
2000
1500
1000
500
0
00
12
00
12
00
12
00
12
00
Hours
Figure 2. (top) PV (given in equation (16)) for two different
values of the dimensionless numerical parameter ˛ . (bottom)
The total energy dissipation UT (time shifted 30 min according to the cross correlation) during the superposed epoch
events.
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TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
1
Correlation Coefficient
0.8
0.6
0.4
0.2
0
−4
−3
−2
−1
0
α
1
2
3
0.8
Figure 3. Superposed correlation analysis between PV (˛)
and the total energy sinks, UT , based on the average values
from the 45 geomagnetic storms. The maximum correlation,
R = 0.84, is found at ˛ 0.5.
4. Events and Methodology
[28] We have identified 45 isolated geomagnetic storms
where 7 of these are severe. A geomagnetic storm is defined
“as an interval of time when a sufficiently intense and longlasting interplanetary convection electric field leads, through
a substantial energization in the magnetosphere-ionosphere
system, to an intensified ring current strong enough to
exceed some key threshold of the quantifying storm time Dst
index” [Gonzalez et al., 1994]. These are identified based on
a criterion that there are no other storms 3 days prior to the
event and no new storms during the recovery phase. Using
these storms, we create an averaged storm which we define
to last 4 days. We have chosen to align the data using the
minimum SYM-H values as the common incident. The duration of the storm is defined to 1 day prior to the minimum
value of SYM-H, interpreted as energy-loading phase, and 3
days after representing energy-unloading phase. In order to
analyze the energy flow for a wide range of conditions, we
have also identified 29 isolated substorms and several long
time periods with consistent solar wind data. For the long
time periods, we present three intervals: 20 days, 4 months,
and our whole data set 13 years.
[29] In order to determine the functional form of the
energy coupling function, we perform a cross-correlation
analysis between the total energy sinks (UT ) in our system
and PV with F() = sin4 ( /2) using the data set described.
Given such a determination, we can analyze the amount of
energy transferred and deposited to the inner magnetosphere
and ionosphere by integrating the energy coupling function
and estimated sinks over a period of time. This allows us to
scale the effective area of interaction such that the energy is
balanced in the system. We also perform an analysis of how
the coupling function performs during a wide range of conditions and time scales, again letting the effective area be a
free parameter.
5. Correlation Analysis
5.1. General Energy Transfer Formula
[30] Using equation (16) with F() = sin4 (/2), constant
area (lCF = const.), and allowing ˛ to vary between –4 and
4, a cross-correlation analysis between the energy sinks, UT
and PV (˛), is performed. We allow both ˛ and the time shift
1997
2002
2009
0.6
0.4
0.2
0
−4
4
−3
−2
−1
0
α
1
2
3
4
Figure 4. Correlation analysis between PV (˛) and the total
energy sinks, UT for three different periods. The correlation
coefficient has its maximum when the dimensionless numerical parameter ˛ 0.5. Approximately, 4% of the OMNI
solar wind data have been linearly interpolated to compute
the correlation.
to vary to maximize the correlation coefficient. In terms of
illustrating the method, Figure 2 shows PV (equation (16))
for ˛ equal to 0.5 and 1 and the total energy dissipation UT =
UR + UJ + UA for the superposed geomagnetic storms. We
note that when ˛ = 1, the resulting functional form is the
(transverse) parameter based on BT .
5.2. Geomagnetic Storms
[31] We perform the correlation analysis based on the
geomagnetic storm events between PV (˛) and UT . For each
storm, the entire storm interval is used to estimate correlation between PV (˛) and UT for various values of ˛ , and
the average coefficient for the 45 storms is plotted for a
given ˛ . Figure 3 shows the result, the x axis represents the
dimensionless numerical parameter ˛ in equation (16), and
the y axis represents the correlation coefficient. We found
the maximum correlation, R = 0.84, when ˛ 0.5. The
energy sinks are time shifted on average 30 min. We also
considered separate shifts for the UR compared to UA and
UJ . However, since the pressure corrected SYM-H inside UR
Deviation From Energy Sinks
Correlation Coefficient
1
1.5
1
Substorms
Storms
Severe Storms
0.5
0
−0.5
−1
0
10
20
30
40
50
60
|Bz| [nT]
Figure 5. Scatterplot between the deviation from energy
sinks and the absolute magnitude of the Bz component which
has been averaged 30 min during maximum solar wind
driving. The energy input is underestimated for substorm
events (red), the energy is balanced for minor to intermediate storms (blue), and the energy input is overestimated for
severe storms (black).
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Deviation From
Energy Sinks
TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
1.4
1.2
1
0.8
0.6
0.4
0.2
−0
−0.2
−0.4
0
2
1
0
2000
0
1000
0
200
5
10
15
20
25
30
35
40
45
0
Events
500
Figure 6. Deviation of the integrated energy input coupling
function from the integrated energy sinks for geomagnetic
storms. Blue filled points represent the energy coupling
function with a dynamic effective area of interaction, while
open red points are for a static effective area of 126 R2E . The
standard deviation (mean deviation) is 0.13 (0.10) and 0.29
(0.18) for the blue and red points, respectively.
often will begin to decline prior to the original SYM-H, the
effect of separate shifts minimizes.
5.3. Long Time Series
[32] Figure 4 shows the correlation analysis for three different years. Even though the shape of the curves differs, the
maximum correlation is found at ˛ 0.5. The functional
form with best correlation during both geomagnetic storms
and long time series is then
Pinput (˛ = 0.5) =
P1/2
f Vx BT
5.4. Effective Area of Interaction
[33] To determine the effective area of interaction 2LM LT ,
we scale Pinput to the energy sinks in the system and require
that
Z
t2
t1
Poutput dt = W(P) [J] = W(UT ) [J]
d)
e)
f)
12
00
12
00
12
00
12
00
Hours
Figure 8. Energy available and transferred from the solar
wind and the various energy dissipations into the inner
magnetosphere and ionosphere. (a) Available solar wind
kinetic energy. (b) Energy coupling function Pstorm given in
equation (23). (c) The total energy dissipated. (d) Global
energy deposition by electron precipitation. (e) Joule heating
of the ionosphere. (f) The ring current injection rate.
[34] The total output is the time-integrated energy sinks in
the system:
Z
t2
W(UT ) [J] =
(20)
UT (t)dt
t1
Deviation from energy sinks =
W(P)
–1
W(UT )
(21)
For the case of the geomagnetic storm events, we determine
the effective area which has the least standard deviation with
respect to the events. We find that the deviation from energy
sinks for the minor to intermediate storms has best statistics
when the effective area 2LM LT = 126 R2E . In Figure 5 we
have shown the deviation from energy sinks using 126 R2E
as the effective area in equation (18). The x axis represents
the 1 h averaged Bz magnitude during the maximum solar
wind driving for each event. Figure 5 shows that the energy
input is underestimated for substorms and overestimated for
t2
Pinput dt =
c)
0
00
(18)
where we have used Vx instead of V, and 2LM LT instead of
lCF , the former due to higher correlation coefficient while the
latter was done by the assumption that the effective area of
interaction is a product of the merging length on the dayside and a transfer region on the tail boundary [Siscoe and
Crooker, 1974; Kan and Akasofu, 1982; Gonzalez, 1990].
Z
b)
where UR , UJ , and UA are defined in section 2.1. We define
the fractional deviation of the integrated energy input W(P)
from the integrated output W(UT ) as
4
2LM LT sin ( /2)
1/2
0
B2 Vx
= T MA sin4 ( /2)2LM LT
0
0
500
a)
800
(19)
t1
600
[1014 J]
where we use (2 LM LT ) as a scaling factor.
Joule Heating
Ring Current Injection
Particle Precipitation
400
150
200
100
0
00
50
0
0
12
00
12
00
12
00
12
00
Hours
10
20
30
40
50
60
Figure 7. Function G(Bz ) (equation (24)) describing the
dynamic effective area of interaction.
Figure 9. Estimated averaged energy sinks and their contribution to the total during the geomagnetic storm events.
The contribution is distributed as follows: Joule heating
47.5%, particle precipitation 23%, and ring current injection
29.5%.
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TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
1500
[1014 J]
a)
1000
500
W(Pstorm)
W(U )
T
W(t) [1014 J]
0
b)
200
100
0
−100
00
12
00
12
00
12
00
12
00
Hours
Figure 10. (top) The average integrated energy sinks
(dashed line) and energy coupling function (solid). (bottom)
The energy budget, W(t), during the averaged geomagnetic
storm.
severe storms. We have interpreted this as that the effective
area should be considered dynamic, and we have found that
the deviation correlates best with the magnitude of the Bz
component.
5.4.1. Geomagnetic Storms
[35] We have chosen to represent the dynamic effective
area using a power fit with two terms. An algorithm finds the
coefficients yielding the best mean and standard deviation of
normalized values using the storm events shown in Figure 5.
[36] In Figure 6 we have shown the deviation from energy
sinks for each of the 45 geomagnetic storms (of which
Events 38–45 are severe storms). The red data points are the
calculated deviation using equation (18) with a static effective area (2LM LT ) of 126 R2E , while the blue filled points are
calculated using a dynamic effective area. The statistics are
considerably better for the dynamic (blue) with a standard
deviation of 0.13 compared to 0.29 for the static area (red).
[37] The dynamic effective area of interaction found to
yield the best statistics is
135
G(Bz ) = 2 LM LT =
5 1022 |Bz |3 + 1
(22)
The energy coupling function for geomagnetic storms can
then be given as
Pstorm =
B2T Vx
135
R2
MA sin4 ( /2)
0
5 1022 |Bz |3 + 1 E
(23)
We have defined the coefficient 5 1022 to have units T–3
in order for the effective area term given in equation (22)
to be dimensionless. Also, we emphasize that this term,
which represents the area where the energy transfer occurs,
is purely empirical and should be considered as a first
approximation to an observed effect. How it relates to
the dimensional analysis [Vasyliunas et al., 1982] will be
discussed.
5.4.2. Long Time Series
[38] When analyzing geomagnetic storms and substorms,
it is important to carefully determine the integration period.
Still it is likely that for some of the events, the magnetotail stores some of the supplied energy, while for
some events, priorly stored energy is released in addition to the energy supplied in the specific event. By
looking at sufficiently long periods, we can investigate
how the magnetotail distributes the energy. By analyzing
several periods ranging from several days to years with consistent solar wind data, we find that the energy coupling
function performs best when the effective area of interaction
is
G(Bz ) = 2 LM LT =
167
5 1022 |Bz |3 + 1
(24)
When considering Figure 5 where the energy transfer was
underestimated for the events with Bz < 10, it is not surprising that we find that we must increase the effective area.
The function G(Bz ) describing the dynamic effective area is
shown in Figure 7. The energy coupling function applicable
for long time periods is then
Pinput =
B2T Vx
167
R2
MA sin4 ( /2)
0
5 1022 |Bz |3 + 1 E
(25)
Thus, the effective area found for long time series is approximately 20% larger than that for geomagnetic storms. This
will be discussed further in section 7.
6. Energy Budget
[39] The energy budget can be established by comparing energy sinks and sources. By assuming a closed system,
we try to determine the spatial and temporal evolution of
the energy flow through the system. The energy sinks, also
called energy output, describe the processes driven by the
energy in the system. The energy source, also called energy
input, is described by a function depending on solar wind
parameters and is a proxy for when and how the magnetosphere couples with the solar wind.
Table 1. The Calculated Time-Integrated Energy Depositions, Energy Coupling Function Pinput , Parameter,
Total Solar Wind Kinetic Energy USW , and Coupling Efficiency for Different Periods and Events
Events
W(UJ )
[1014 J] (%)
W(UA )
[1014 J] (%)
W(UR )
[1014 J] (%)
W(UT )
[1014 J]
W(P)
[1014 J]
W()
[1014 J]
W(USW )
[1014 J]
CE
(%)
Substorm
Storma
20 days
4 months
1997
2003
1997–2010
28.5 (54.5)
655 (47.5)
1,394 (54)
13,800 (55)
30,300 (52)
53,200 (54)
480,000 (54)
17.5 (33.5)
320 (23)
878 (34)
7,610 (30)
19,100 (33)
27,900 (28)
270,000 (31)
6.4 (12)
406 (29)
299 (12)
3,680 (15)
8,600 (15)
17,300 (18)
136,000 (15)
52.3
1,381
2,571
25,150
58,000
98,400
889,000
48
1,394
2,582
25,210
59,300
106,400
864,000
27
2,018
1,986
18,580
35,900
94,600
668,000
7,898
1.69 105
4.69 105
0.386 107
1.04 107
1.64 107
14.9 107
0.66
0.82
0.55
0.65
0.56
0.6
0.6
a
We have used Pstorm for the energy input during the storm events. Using Pinput would increase the energy input estimate by
approximately 20% to Pinput = 1700 1014 J.
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TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
100
a)
Deviation From
Energy Sinks
SME [nT] Sym−H [nT] W(t) [1014 J]
200
0
−100
20
0
−20
−40
1000
b)
c)
500
Date [UT]
Figure 11. The energy balance calculated using (a) Pinput ,
(b) SYM-H index, and (c) SME index in the period 1 to 21
April 1998. The calculated time-integrated values associated
with this event are presented in Table 1.
[40] Such an analysis can help to understand the energy
flow in the whole solar wind-magnetosphere-ionosphere
system and determine whether the sources and sinks balance
in the system.
[41] To calculate the energy balance, we calculate the
time-integrated energy input and output as a function of
time:
t2
W(t) =
(26)
P(t) – UT (t) dt
t1
where UT = UA + UJ + UR , and P(t) is the energy coupling
function.
6.1. Geomagnetic Storms
[42] We have chosen to use the Pstorm parameter when presenting the energy balance for geomagnetic storms. Figure 8
shows the available solar wind energy, the energy coupling function Pstorm , and the energy deposition derived from
SuperMAG and Kyoto. The energy sinks, UJ , UR , and UA ,
during the superposed geomagnetic storms give the follow-
W(t) [1018]
0.15
0.05
0
2.5
[1018 J]
a)
0.1
−0.05
b)
2
1.5
1
W(Pinput)
0.5
0
04/01
W(U )
T
05/01
06/01
07/01
P
storm
ε parameter
5
10
15
20
25
30
35
40
45
Events
0
04/01 04/03 04/05 04/07 04/09 04/11 04/13 04/15 04/17 04/19 04/21
Z
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
0
08/01
Date [UT]
Figure 12. (a) The energy balance calculated and (b) the
time-integrated energy coupling function Pinput and UT , in
the period 1 April to 1 August 1998. The integrated energy
input in this period W(Pinput ) = 2.5219 1018 J, the total
deposited energy W(UT ) = 2.5156 1018 J, and other
calculated values are summarized in Table 1.
Figure 13. Comparing the deviation from energy sinks for
the coupling function Pstorm (blue) and the parameter (red)
during the geomagnetic storm events.
ing contributions to the total (UT ), 47.5%, 29.5%, and 23%,
respectively, see Figure 9.
[43] Figure 10 shows the energy budget during an average geomagnetic storm, calculated using equation (26). The
average energy input for the average geomagnetic storm
is found to be W(Pstorm ) = 0.139 1018 J, and the timeintegrated energy depositions are W(UT ) = 0.138 1018 J,
see Table 1.
6.2. Long Time Series
[44] When analyzing longer periods, we use Pinput and
will present three different intervals, 20 days, 4 months,
and 13 years. The periods analyzed contain both storms and
substorms as well as quiet periods with no obvious energy
transfer. Figure 11 shows the energy budget, SYM-H, and
SME for a period of 20 days. The purpose of the figure is to
show how the energy coupling performs during a relatively
quiet period. Still there are several substorms and minor
storms; however, the energy coupling function estimates all
the events well, as is evident in the energy budget.
[45] In this period, the 0.258 1018 J was transferred from
the solar wind to the magnetosphere and 0.257 1018 J was
deposited in the inner magnetosphere and ionosphere. The
energy sinks, UJ , UA , and UR , during this period give the
following contribution to the total (W(UT )): 55%, 33%, and
12%, respectively. These values and more are summarized
in Table 1.
[46] Figure 12a shows the energy budget, while Figure 12b
shows the time-integrated energy coupling function Pinput
and total energy dissipation UT during a period of 4 months.
In this period, there is a severe geomagnetic storm (occurring 4 May 1998), where the magnitude of IMF Bz reaches
values below –35 nT. We observe the huge amount of energy
driven into the system during this storm and then dissipated
at a somewhat lower rate. The saturation term (G(Bz )) in
the energy coupling function is crucial for the success of
the energy budget during such events. The calculated timeintegrated energy input, energy depositions, parameter,
solar wind kinetic energy, and coupling efficiency for this
event, as well as for the years 1997 (close to solar minimum) and 2003 (close to solar maximum) are summarized
in Table 1.
6.3. Coupling Efficiency
[47] Another important parameter is the coupling efficiency, which is how much of the available solar wind
5667
TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
5
a)
[1018 J]
0
5
−10
−15
−20
−25
[1018 J]
80
b)
60
40
W(Pinput)
20
W(U )
T
0
1998
2000
2002
2004
2006
2008
2010
better. However, it is evident that Pstorm performs better for
the various events and that it handles the relevant properties
of the energy input more accurately.
[50] Figure 14 shows how the parameter performs for
long time series and a comparison with the energy coupling
function Pinput . Figure 14a shows the energy balance W(t)
(equation (26)), and Figure 14b shows the time-integrated
energy coupling function Pinput and total energy sinks UT .
During this period, the parameter underestimates the
energy input by approximately 33% while Pinput underestimates by 3%. During this 13 year period, the parameter
underestimates the energy input and should be scaled by
a factor of 1.3. However, we note that during the severe
storms, identified by the arrows in Figure 14, the parameter overestimates greatly; thus, it is likely that the scaling
factor is somewhat higher during quieter periods.
Date [UT]
Figure 14. (a) The energy balance, the black line, is calculated using Pinput , while the red line is the parameter in the
period 1997–2010. The arrows indicate severe storms where
the parameter clearly overestimates the energy input. (b)
The integrated energy input and output.
kinetic energy that actually penetrates the magnetosphere.
The coupling efficiency is defined as CE = W(UT )/W(USW ).
Østgaard et al. [2002b] found the coupling efficiency of
the solar wind kinetic power to the magnetosphere to be
from 0.3 to 0.8%. Stern [1984] estimated it to be roughly
1%, while Lu et al. [1998] suggest as much as 4%. We
note that these results depend on the magnetospheric cross
section used. During a geomagnetic storm, we find that CE
on average is 0.8%, and it reaches values above 1.3% during
maximum solar wind driving. For the period 1997–2010, we
find the average value of 0.6%. Our results are summarized
in Table 1 and are in general agreement with Østgaard et al.
[2002b] and Stern [1984].
6.4. A Comparison With the Parameter
[48] To our knowledge, as the parameter is the only
energy coupling function scaled to the energy sinks, we find
it reasonable to present a comparison. The parameter was
originally determined to analyze geomagnetic storms; for
this reason, it should be compared to our Pstorm . The functional form of the coupling functions is similar yet there
are essential differences. Pstorm uses the x component of the
solar wind velocity Vx and the transverse magnetic field
component compared to the parameter’s VB. Pstorm has an
Alfvén Mach number dependence, as will be discussed later.
Also, we let the effective area of interaction to be dynamic
and dependent to the first order on the magnitude of the Bz
component. So for all practical reasons, only the angular
dependence in the two functional forms is common.
[49] Figure 13 shows the deviation from energy sinks
for both Pstorm and the parameter for the 45 geomagnetic
storm events. On average, the parameter overestimates the
energy input for geomagnetic storms. Especially during the
severe storms where it overestimates the energy input by as
much as 350% (Event 45 in Figure 13). We note that the parameter was originally scaled using different estimations
of the energy sinks; thus, we should not expect it to perform
7. Discussion
[51] In section 2 we mentioned uncertainties related to
the estimation of the energy sinks. Regarding Joule heating, there is an ambiguity in the community about how it
should be estimated. Palmroth et al. [2005] review different
methods and techniques and present a comparison of Joule
heating estimates from the Ahn et al. [1983] proxy, AMIE,
Super Dual Auroral Radar Network, Polar satellite measurements, and GUMICS-4. During the event studied, Palmroth
et al. [2005] argue that the AMIE estimate is too high during peak but consistent with what the authors consider to
be the “true” Joule heating toward the end of the substorm.
The Ahn et al. [1983] proxy is regarded as consistent, however, it is a factor 2 higher during peak. The “true” Joule
heating, which is determined using MHD simulations multiplied by a factor of 10, and theoretical arguments coincide
well with the global observations [see Palmroth et al., 2005,
Table 1] and appear to be less than suggested by, for example, the AE proxies [Palmroth et al., 2005]. We acknowledge
the uncertainty in the Joule heating estimation using AE
proxies in this study and that even using the same data the
different methods can yield divergent estimates. However,
we have used the experimental estimates that in our opinion
reflect most accurate the amount of Joule heating dissipated
in the ionosphere. Also, we note that in the MHD simulations
[Palmroth et al., 2004], both the temporal and spatial variations are well correlated with the empirical proxies, such
that it should not affect our correlation analysis significantly.
We have used values corresponding to solstice, UJ = (0.33 +
0.21)AE + 1.8 (summer + winter) [Østgaard et al., 2002b].
These may differ in the cycle of seasons. Ahn et al. [1983]
suggested UJ = 0.23AE for one hemisphere during equinox,
which we consider reasonable, less than for summer and
higher than during winter. Thus, for equinox, Joule heating
could be expressed as UJ = (0.23 + 0.23)AE = 0.46AE which
represents a deviation of about 15%. The consequence of
this uncertainty in the estimated energy sinks will slightly
affect the energy coupling function. Even though the functional form will not change, the effective area of interaction
will differ. By determining the effective area for geomagnetic storms using an expression for the Joule heating which
is scaled ˙15%, we find that this results in increase/decrease
of about 7% for the effective area. That is,
5668
Pstorm =
B2T Vx
135 ˙ 9
R2
MA sin4 ( /2)
0
5 1022 |Bz |3 + 1 E
TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
with corresponding percentage uncertainties in Pinput . Considering the extremes where UJ is off by as much as a
factor of 0.5 or 2, this would imply a deviation of the effective area by –30 or +65, respectively. These extremes will
be discussed further below. Similarly regarding the particle
precipitation, we cannot discard seasonal variations. Our
results rely on the study by Østgaard et al. [2002b] where
the analyzed events occurred during solstice. Compared to
previous studies, e.g., the values used by Akasofu [1981]
only provided 30% of the UA . As an example, applying an
uncertainty of 50% in UA (equation (6)) will result in a 10%
change in the effective area.
[52] For the case of uncertainties in the estimation of the
ring current injection rate (equation (9)), the consequence
of choosing a dynamic decay rate [O’Brien and McPherron,
2000] will affect the scaling. During Event 7 which has
a minimum SYM-H of about –120 nT, the calculated integrated energy deposition UR is approximately 40% lower
when using the dynamic decay rate. For more severe storms,
this would result in an even larger difference, and for weaker
storms, the difference would diminish. For this particular
storm, the deviation in the effective area in our energy
coupling functions would be approximately 6%.
[53] As discussed, it has been proposed that both UJ and
UA could be off by some factor 2. If both these processes
were overestimated by a factor of 2, it would imply that UR
is the main dissipation process in the system (at least during
storm time): UJ + UA UR . In this extreme, the functional
form should be reconsidered. In the more likely situation
where UJ + UA > UR , our functional form will hold and only
the scaling must be changed.
[54] We also would like to comment on the angular dependence, sinˇ ( /2). The exponent value of 4 was chosen based
on previous studies [Gonzalez and Mozer, 1974; Akasofu,
1981; Kan and Akasofu, 1982], a correlation analysis where
ˇ varied from 0 to 5, and trial and error on different events.
The correlation analysis showed that the correlation coefficient did not change significantly for ˇ > 1. In a trial-based
analysis, we considered our 45 geomagnetic storms (see
Figure 6) and an algorithm calculated the standard deviation of the 45 events for different values of ˇ . In this
case as well, the value of ˇ did not significantly affect the
results. The standard deviation improved as ˇ increased,
however, only on the second to third decimal. Choosing a
higher ˇ value implies a larger effective area in order to balance the energy. Thus, changing the ˇ requires scaling the
effective area, and it is possible that these values are dependent. Gonzalez and Mozer [1974] discussed how the compression ratio in the magnetosheath could affect the value
of ˇ .
[55] We have found two energy coupling functions. Pstorm
is scaled to fit geomagnetic storms, while Pinput is applicable for long periods. It is well known that the magnetotail
stores magnetic energy; therefore, it may not be surprising
that these differ. We interpret Pstorm as the energy directly
deposited in the inner magnetosphere and ionosphere
[Rostoker et al., 1988; Baker et al., 1997; Boakes et al.,
2011]. Comparing Pinput and Pstorm would imply that approximately 20% of the transferred energy is stored in the
magnetotail during an average geomagnetic storm. The
dayside flux rate, ˆD , could exceed, equal, or be less than
the rate of closing flux on the nightside, ˆN . The former
defines how much flux is transported to the tail, while the
latter represents how much of the available magnetic flux
is transported from the magnetotail to the inner magnetosphere and ionosphere. Tanskanen et al. [2005] found that
during the substorm expansion phase, ˆN usually exceeded
ˆD ; previously stored flux is then released in addition to the
flux which is directly transported.
[56] As previously defined, the effective area of interaction is a product of the geo-effective length, LM , and the
transfer length, LT . We found that the area is dynamic and
varies to the first order on the magnitude of the Bz cubed.
In terms of dimensional analysis presented by Vasyliunas
et al. [1982] (see equation (16)), the fundamental dimensions may only appear in combinations having appropriate
dimensions. Thus, in order to be consistent with this analogy,
Bz should be multiplied with the inverse dynamic pressure
(see equation (17)). This, however, significantly decreased
the performance. The part of the coupling function that handles the actual energy transfer/conversion is consistent with
the dimensional analysis. However, the effective area representing the area at which the actual energy transfer occurs,
rather than the actual energy transfer, is empirical and not
consistent with dimensional analysis. We also note that the
empirically determined numerical factor 5 1022 is very
close to the Earth’s magnetic dipole moment ME = 7.94 1022 which is probably a coincidence.
[57] Another question arises: How does the area we have
estimated correspond to the position of reconnection in the
tail? In the magnetosheath, the solar wind is channeled
around Earth’s magnetosphere, and only a fraction of the
solar wind interacts with the magnetopause. The limited
extent of the solar wind flow that participates in the dayside
reconnection process is the geo-effective length, LM [Lopez
et al., 2011]. Reiff et al. [1981] estimated LM to be roughly
0.1 to 0.2 of the magnetosphere width. More recent studies
argued that LM is in the range of 5–8 RE , which was later
adjusted to 3 RE [Milan, 2004; Milan et al., 2004, 2008,
2012]. By assuming a merging length of LM 3 RE , we can
estimate the transfer length LT to be approximately 28 RE
during a geomagnetic storm. For a severe magnetic storm
with an average IMF Bz of 20 nT, LT would decrease
to about 20 RE . This corresponds well to our assumption
that the transfer length is bounded to the position of the tail
reconnection [Nagai et al., 2005].
[58] As mentioned in the introduction, we will now discuss how our semiempirical result can be interpreted in view
of the Maxwell stress tensor. Considering equation (15),
the integral of Bn V can be related to upstream solar wind
parameters, and the reconnection rate at the dayside magnetopause is ultimately responsible for the normal component
Bn , and thus ultimately the tangential stress. In absence of
tangential stress, the magnetopause and tail currents are dissipationless currents on the magnetopause surface; however,
when a normal component exists, these currents can influence the flow in the dayside and tail regions, respectively
[Russell, 2000]. Applying
H rewrite
H this consideration, we can
E 0 d = Vx BT LM dlBt /0
equation (15) to P = Bn BEt V/
[see Vasyliunas et al., 1982]. In the familiar laws of electric circuit theory, it is evident that P represents the cross-tail
electric field times the total current. From our semiempirical coupling function (equation (18)), the total current
or the tangential component of the magnetic stress tensor
5669
TENFJORD AND ØSTGAARD: ENERGY TRANSFER AND FLOW
R
p
is given by dlBt /0 = LT 0 Pf /0 = LT BT MA /0 [cf.
Akasofu and Perreault, 1978; Kan and Lee, 1979; Gonzalez
and Gonzalez, 1981; Vasyliunas et al., 1982].
[59] The role of the bow shock in magnetospheric dynamics has generally been unappreciated in the energy coupling
between the solar wind and magnetosphere [Lopez et al.,
2011]. How the solar wind interacts with the magnetopause
is shock dependent [Lavraud and Borovsky, 2008]. These
properties are Mach number dependent. The coupling occurs
at the magnetopause; hence, the characteristics of the magnetosheath properties should be included in the coupling
function. It is of our understanding that the Alfvén Mach
dependence in the energy coupling function is contributory to capturing the distorted upstream properties of the
solar wind in the magnetosheath. The Alfvén Mach number
describes the square ratio of kinetic energy flux to the electromagnetic energy flux and reflects, among other things, the
position of the bow shock and the amount of compression in
the magnetosheath. For the high Mach number regime, the
bow shock compression ratio is close to the theoretical maximum of 4 [Lopez et al., 2004]. The Mach number parameter
in the energy coupling function will contribute most during such conditions, influencing the energy transfer rate. For
the low Mach number regime (magnetosheath is magnetically dominated), the magnetosheath flow is controlled by
the Lorentz force, J B. In this regime, the coupling function dependence on the Mach number acts to saturate the
energy input [Lopez et al., 2011; Lavraud and Borovsky,
2008]. However, Lopez et al. [2004] showed that under
these conditions, the compression ratio is strongly dependent on variation in the pressure. Higher density results in
larger compression ratio (higher Mach number) resulting in
a stronger magnetic field applied to the magnetosphere. The
compression ratio also influences the amount of draping,
which again could increase the magnetic stress on the magnetopause. We also notice that by using the magnetosheath
model by Crooker et al. [1979], thepmagnetosheath field
strength can be expressed as BSH / MA BT . Inserting this
in our energy coupling function would then give a funcV B2
tional form similar to the parameter: x0SH sin4 ( /2)2LM LT
(if 2LT LM = l20 ). Our semiempirical energy coupling function
coincides with several arguments on how the magnetosheath
behaves during different regimes [Lopez et al., 2004, 2011;
Borovsky, 2008].
8. Conclusions
[60] In this paper we have presented two new dynamic
energy coupling functions, derived using 5 min OMNI and
SuperMAG data. The 5 min resolution is considered to be
the optimal and characteristic time scale [Tsyganenko, 2013].
Pstorm , derived using 45 geomagnetic storms, is applicable when studying geomagnetic storms but underestimates
the energy input for longer time series. It is given as
(equation (23))
Pstorm =
B2T Vx
135
R2
MA sin4 ( /2)
0
5 1022 |Bz |3 + 1 E
q
p
P
where BT = B2y + B2z and MA = BT0 f . By studying longer
time series ranging from days up to 13 years, we found
that the effective area of interaction had to be increased
by approximately 20%. The energy coupling function Pinput ,
which is applicable for longer time series, is given as
(equation (25))
Pinput =
B2T Vx
167
R2
MA sin4 ( /2)
0
5 1022 |Bz |3 + 1 E
We have argued that the difference between Pstorm and Pinput
could be explained by the magnetospheric response to the
accumulation of energy in the tail. We also note that defining the geomagnetic storm to have a longer duration would
diminish the difference. That is to say that 20% of the
energy accumulated in the tail is released subsequently of
the defined storm period.
[61] Throughout the paper we have attempted to relate the
terms from our semiempirical analysis to theoretical considerations. The Alfvén Mach dependence in our energy
coupling function improves the performance of the energy
coupling function by increasing the transfer rate for high
Mach upstream solar wind and mitigates the energy input
during low Mach conditions.
[62] In Table 1 we have listed the time-integrated energy
depositions, energy coupling functions Pinput , Pstorm , the parameter, total solar wind kinetic energy USW , and coupling efficiencies for all the events discussed in the paper.
In general, the parameter underestimates the energy input.
This is evident when we look at long time series. Only
during geomagnetic storms does the parameter overestimate the input, but far from enough to compensate
for the general underestimate. Thus, for all conditions,
we find the dynamic energy coupling function presented
in this paper to perform better than . We believe this
new dynamic coupling function can provide new insight
to which conditions are more or less favorable for energy
transfer and also can be used to study spatiotemporal variations in solar activity, the evolution of energy deposition
for long time scales, and how the magnetosheath properties affect the efficiency of the energy transfer among
other things.
[63] Acknowledgments. This study was funded by Research Council of Norway under contract 223252/F50 to the Birkeland Centre for
Space Science as well as contract 216872/F50. We acknowledge the use
of NASA/GSFC’s Space Physics Data Facility’s OMNIWeb service and
OMNI data. For the ground magnetometer data, we gratefully acknowledge
the following: INTERMAGNET; USGS, Jeffrey J. Love; Danish Meteorological Institute; CARISMA, Ian Mann; CANMOS; the S-RAMP Database,
K. Yumoto and K. Shiokawa; the SPIDR database; AARI, Oleg Troshichev;
the MACCS program, M. Engebretson, Geomagnetism Unit of the Geological Survey of Canada; GIMA; MEASURE, UCLA IGPP, and Florida
Institute of Technology; SAMBA, Eftyhia Zesta; 210 Chain, K. Yumoto;
SAMNET, Farideh Honary; the institutes who maintain the IMAGE magnetometer array, Eija Tanskanen; AUTUMN, Martin Conners; Greenland
magnetometers operated by DTU Space; South Pole and McMurdo magnetometer, Louis J. Lanzarotti and Alan T. Weatherwax; ICESTAR; RapidMag; PENGUIn; British Antarctic Survey; McMAC, Peter Chi; BGS,
Susan Macmillan; Pushkov Institute of Terrestrial Magnetism, Ionosphere
and Radio Wave Propagation (IZMIRAN); and SuperMAG, Jesper W.
Gjerloev.
[64] Philippa Browning thanks Joseph Borovsky and an anonymous
reviewer for their assistance in evaluating this paper.
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